I am an engineer-turned-school-maths teacher....
I love to see the sparkles of understanding in the eyes of my students.... It is really exciting to see I was part of this enlightenment process.... I find myself both inspired and inspiring!
I love doing math with children & sharing my love of math and children with parents and teachers... Check the websites www.about.me/rupesh.gesota and www.supportmentor.weebly.com to know more about me & my math-adventures...

(It was about how beautifully a visually challenged student had solved a problem in one of the non-conventional ways :)

I remember I had talked to you about the deal that I will post a story in response to your 2 stories, but am sorry -- I couldn't resist my urge of sharing with you all a very interesting approach of doing multiplication, as discussed by one of my students today. His name is Arjun and he is part of the bunch of 8th std. students of a municipal school with whom I have recently started working with, voluntarily after their school hours.

Yesterday, we had stumbled upon a problem where we felt the need for doing the multiplication as a part of its intermediate step. I don't remember the numbers but it was a 2-digit no. multiplied by 1-digit number. The numbers were quite easy, still the students were unable to do i mentally and wanted to resort to pen-paper. I insisted for mental maths and after some discussions, the team could Discover the meaning of multiplication and gradually, they even started doing these 1-d x 2-d problems Mentally, and with Understanding and Not by using the conventional standard procedure of multiplication (something that I and perhaps most of us would have studied/used, and probably even without understanding, in our school days :)

{I am not elaborating these strategies in this post as I assume most of the teachers must be aware of these understanding-based approaches.... one can even google out)

It was the end of the 2-hour session and so they requested me to give them some problems on multiplication as assignment, so that they can practice this new approach (Understanding-based) which they had started loving a lot. So I gave them about 10 problems.

So today, when we were done with our 10-minute math warm-up, we started discussing the assignment problems. It was so fulfilling to notice that each of these students had solved each of these problems in more than 1 way, by Understanding and at times even with the application of Logic. For example: For solving 38x5, they first worked out 38x10 which equals 380 and this was halved to get 190 as the required solution. And there were many more beautiful strategies.....which can easily drop the jaws of many of us ;)

But in this post, I want to specifically share an innovative approach figured out by Arjun to the problem: 29x8 (of course, his approach is actually an off-shoot of the understanding of the previous strategies, but the way he has remarkably applied this strategy to solve this problem -- this is something that blew off my mind !! )

He says:

Lets consider (29 x 8) as (30 x 10) which yields the product 300.

But we could have also considered (29 x 8) as (30 x 8) rather than (30 x 10) ... thus we have taken a surplus of (30 x 2) i.e. 60

So lets subtract this 60 from 300 which gives us 240.......which is nothing but (30 x 8).

But now, we wanted to calculate (29 x 8) and not (30 x 8)....... Thus we have further taken the surplus of (1 x 8) which is 8

So we need to subtract this 8 from 240 which gives 232, the solution to the given problem.

I want to make an honest confession, that when he verbally and quickly described these steps at the first time, it went off my head straight-away. It was only when his peer, Suvarna, joined him and re-framed this approach, a bit slowly, I could fathom this strategy...

I was just taken away by the levels of understanding they had transcended to. I was so impressed that I challenged them to solve another problem using the same approach :))

They did not even take a minute to shoot off the approach and solution. Check the snap attached and see if you can comprehend their understanding.

But while they were enjoying and working on these problems using various multiple approaches (based on their conceptual understanding), thoughts flew by my mind... to ask the teachers and parents in this group ---

Should students be encouraged this way to think independently and discover their own approaches to solve the problems?